Computational Design for Digitally Fabricated 3D Inductive Power Transfer Coils

Wireless power transfer (WPT) has attracted a wide range of interests in the past decade [1]. Among different techniques, inductive power transfer (IPT) is a popular mechanism in near-field WPT, and it was adopted for many applications, e.g., the quite interesting (QI) standard [2]. In an IPT system, a pair of 2D planar spiral coils, one or two layers each, are often used. The power that the system can transmit and the efficiency of the transmission highly depend on the geometric shapes, the materials, the relative positions of the transmitter and the receiver coils. For instance, large misalignment in the lateral and/or increased distance between coils in the axial directions may lead to a significant reduction of the coupling factor of an IPT system [3].

The geometric design methods of 2D planar coils have been well developed in the past decades. However, the forms of products do not always contain a planar surface for embedding a 2D coil. For instance, in personalized product designs, the shape of the product should adapt to certain part(s) of the body, such as wearable electronics [4], stretchable electrodes [5], smart skin [6,7], and microhairy sensor [8]. Embedding a planar coil in such a design may require additional space and/or create large lateral misalignments between the transmitter and the receiver coils. Especially when the receiver coil is small, a significant loss of magnetic flux during the power transfer may happen due to minor misalignments, which may result in an underpowered system [9]. Recent development in soft robotics [10] and electronic skins [11] post even more challenges on integrating IPT in the design as the geometric shapes of these products contain many freeform surfaces, and they may change overtime to achieve different functions.

Technology advancement offers new ways in designing and manufacturing coils for IPT in 3D, such as 3D printing [12,13], stamp printing [14], cartan transfer printing [15], etc. For instance, Jeong et al. [16] integrated a IPT system in a smartwatch strap with flexible 2D coils and magnetic field shielding materials. Researchers also developed new inductors [17], sensors [18,19], etc. using 3D printed electronics. However, regarding coils for IPT, most of the designs were achieved by simply projecting, or morphing, a 2D spiral/rectangular pattern to a 3D surface(s). The spatial relations between the transmitter and receiver coils, the distances between each turn of a coil, and the influence of 3D shapes on the electromagnetic field of the IPT system were not fully considered in the design of these coils.

In this paper, we address this gap by introducing a new approach for designing the geometric shapes of 3D coils for IPT. The scientific contribution is that in given region of interests (ROIs), we embed both geometric and electromagnetic constraints in the design of the smallest possible pair of 3D coils to achieve transmitting the desired power. The proposed approach was verified by FEM simulations and experiments on digitally fabricated coils.

This paper is based on earlier work [20] published in ASME IDETC/CIE 2021 conference. The remainder of the paper is arranged as follows: in Sec. 2, a brief literature review was given on different aspects regarding coil design and manufacturing for IPT. Section 3 introduces the proposed approach and in Sec. 4, we present three cases to compare the calculation, the simulation, and the experiment results for verifying the approach. Finally, a short conclusion is presented.

IPT can be used in a variety of products and therefore coils will have to be designed with different shapes and using different materials, e.g., a pair of cylindrical coils with magnet wires is often used for charging the toothbrush [21]. Among different shapes, a pair of 2D planar circular, or rectangular, spiral coils is the most common pattern(s) used in an IPT system. The advantage of using planar circular coils is that a pair of them has higher self- and mutual-inductances than square shaped coils with the precondition that the length of conductors is the same [22]. However, in many designs, the design space of the coil is often a rectangular shape. Using a rectangular spiral coil may achieve a larger inductance than using the circular spiral coil given a defined rectangular design space [23]. Coils in other shapes, e.g., elliptical coils, are also reported in the literature for specific applications [24,25]. No matter which shape is used in an IPT system; coils are often winded in the convex and close-packed forms for reducing the volume and enlarging the self- (and/or mutual) inductance of the coil(s).

Here, the quality factors Q₁ = 2πfL₁/R₁ and Q₂ = 2πfL₂/R₂ are the figure-of-merit of the transmitter and the receiver coils, respectively [27]. L₁ and L₂ are the self-inductance of these two coils, and R₁, R₂, and R_L are the resistances of the transmitter coil, the receiver coil, and the load, respectively [3]. The coupling factor between two coils $K12=M12/L1L2$⁠, where M₁₂ is the mutual-inductance between the transmitter and the receiver coils.

Given the design of a pair of coils for an IPT system, according to these definitions, the geometric shapes of the coils might influence the self-inductances and the resistance of these two coils. The relative position might influence the mutual-inductances between them and the efficiency of the system, e.g., the angular, lateral, and axial misalignments of these two coils often lead to a significant loss of power to be transmitted. In general, for transmitting a given amount of power using IPT, a pair of well aligned coils with large self-inductances and lower resistances is preferred for minimizing the geometric sizes of the coils in the design space, and therefore saving material and manufacturing cost of the coils as well.

Coils are often wound using litz wires [28]. With the development of printed electronics, printable conductive inks have been used to replace conventional conductors in many IPT applications for saving the cost and/or reducing the size while providing enough power for the system [29]. 2D printed electronics, i.e., printing a single layer or multiple layers of silver ink on the substrate, was first introduced for designing low power IPT systems [16]. Conformal printing is often used for translating those 2D designs to 3D application based on the flexibility of the conductive ink and the substrate. In conformal printing, electronic structures are first fabricated on a flat flexible substrate by inkjet printing or screen printing; the print is then deformed to the target (curved) surface using different manufacturing methods [30]. For instance, Harnois et al. [31] fabricated frequency selective surface structures on a 2D planar polyethylene terephthalate substrate. The 2D surface was then bonded onto a 3D surface. Devaraj and Malhotra [32] proposed a new process that can transform flat printed circuits on thermoplastic sheets into freeform interconnect–polymer assemblies and integrated them with a desired 3D surface. A number of studies have confirmed that thermoforming technology can also be used for deforming the printed large-area interconnect circuits towards the desired shape [33]. Similarly, by combining the in-mold decoration technique and 3D printed electronics, in-mold electronics was also widely adopted by industries for manufacturing functional electronics [34].

Direct ink writing is another way of digitally fabricating electrical circuits in 3D [15]. For instance, Zhou et al. [35] fabricated a broad array of 3D radio frequency passive components (e.g., inductors, capacitors, etc.) for chip-scale radio frequency electronics. Using 3D printed electronics, Boekraad [36] succeeded in printing a coil onto a curved surface using a set of non-planar toolpaths. However, it might be difficult to generalize this approach to other applications, especially for printing on freeform surfaces. Zhu et al. [37] developed an on-site 3D electronics printing technique to print sensors as well as a coil for WPT on human hands. Recently, Hou et al. [3,38] mapped 2D rectangular/circular spiral patterns to arbitrary freeform surfaces for generating 3D coils and validated the effectiveness and efficiency of using those coils in the IPT system based on 3D printed electronics.

When using printed electronics for IPT, coils are often designed in 2D using the circular/rectangular spiral patterns with the step size (distance between turns) as the sum of the width of the printing trace and a specified clearance between each turn of the coil. However, in the use of the IPT system, those 2D designs are either deformed by the users following particular usage scenarios, e.g., flexible electronics [16], or mapped to a 3D shape using a specific type of mapping [38]. Either might change the geometry and/or the relative position of (each turn of) the coils, leading to a decrease in performance of the IPT system. For instance, the distances between different turns of a coil might change by projecting a 2D design on a 3D curved surface [39]. This might waste the design space or result in a potential risk of short circuits between turns. Another potential issue is that a convex 2D coil might become concave after mapping it to 3D, which will decrease the performance of the IPT system.

Designing coils in 3D can avoid those problems. In the area of computer aided design (CAD), researchers developed many algorithms that can be used in 3D coil design. For instance, the fast marching [40] and the heatmap [41] are two typical algorithms to generate concentric patterns regarding the geodesic distance on the freeform surface. Another example is a new type of five-axis spiral inspection path generation strategy which is able to create a uniform toolpath regarding the geodesic distance [42]. The designed 3D coils can be manufactured using the direct ink writing technique in 3D or using the conformal printing technique by mapping the 3D design to 2D, e.g., by the least-square conformal map (LSCM) [43–45]. However, extensive literature study did not reveal existing research works that integrate the electromagnetic requirements of coils for designing and manufacturing IPT system on a freeform 3D surface.

Given a desired amount of power to be transferred via the IPT system, we propose a 3D coils design approach as Fig. 1. In the first step, two ROIs are specified on two surfaces where the IPT system is supposed to be integrated. A transmitter coil is then generated at an initial position either specified by the designer or at the center of a surface. At first, the size of this coil will only be limited by the boundary of the ROI, i.e., an excessively large transmitter coil will be generated in step 2. Then, a random seed for the receiver coil on the corresponding surface is specified. Using this seed, we grow the receiver coil turn by turn until the size that the system is able to transmit the desired power. The position of the seed of the receiver coil is further optimized until the maximal efficiency is achieved as step 3. In step 4, we flip the order and used the receiver coil to optimize the position and the shape of the transmitter coil based on the desired power to be transferred. Steps 3 and 4 iterate until the changes of the geometric centers of both coils (step 5) are less than a threshold. In the following, details of the above steps are explained.

Using C₀(u) as the basis of the inner turn of the transmitter coil, the basis of the next turn C₁(u) of the coil is computed using the geodesic offset with the same offset value. Here, the shapes of the C₀(u) and C₁(u) may influence the performance of the I system. For instance, the direction of the electromagnetic fields of a coil follows the right-hand rule regarding the direction of the current that flows through it, and in the concave area, the current might generate “negative” electromagnetic fields which will result in a decreased “total” electromagnetic field. Therefore, concave regions of C₀(u) and C₁(u) should be avoided.

After Sprial(θ) is acquired, we project the Sprial(θ) and C_pc1 back to the S_T(u, v), where the projected Sprial(θ) is the extension, i.e., a new turn, of the transmitted coil and the projected C_pc1 is the basis for the next new turn. This process iterates until the boundary of ROI is reached. Finally, the transmitter coil c_t(u) is constructed based on a piecewise curve where each piece is the projected Sprial(θ) regarding this turn. In this case, the transmitter coil is generated with an excessively large size as the starting point of the optimization process as shown in Fig. 4.

Given a starting position S_R(a, b), a receiver coil c_r(u) can be generated in S_R(u, v) using the same strategy as described in the previous section with an exception, which is the power to be transmitted. For designing a 3D coil with the minimal size, the desired power to be transferred (or with an additional threshold) will be used as the upper limit, i.e., the extension of the coil will stop when such a limit is reached. Equation (2) defines the maximal amount of power that the system can transfer. In the equation, it can be found that this limit is influenced by the mutual- and self-inductances of the coils, the resistance of the coil, and the resistance of the load.

Based on Eqs. (12) and (14), a receiver which is able to transfer the desired power with maximal efficiency can be generated. Figure 5(a) presents the generated optimal receiver coil regarding the transmitter coil. It can be found that though the transmitter coil is large, the size of the receiver coil is much smaller, as it is limited by the amount of power to be transferred.

With the generated receiver coil, following the proposed approach, we fix the receiver coil and use the same strategy as Sec. 3.3 to find an optimal transmitter coil. This optimization process iterates until in the kth iteration, both the distance between the center of the receiver coil $SRk(a,b)$ and the optimized $SRk+1(a,b)$ using Eq. (14) and the distance between the transmitter coil $STk(a,b)$ and the optimized $STk+1(a,b)$ using Eq. (14) are less than a threshold, e.g., 0.1 mm. In Figs. 5(b) and 5(c), several steps in such iterative optimization process are presented.

In this section, we present the results of three experiments to verify the proposed approach. In the first experiment, the results of the calculation, the simulation, and the measurements on a prototype are compared. Then we explore the possibility of designing the IPT system for a shaver by comparing the calculation and simulation results. In the third experiment, using the proposed methods, we printed a coil on the liner of a hand splint for providing power to measure physiological parameters.

The proposed design approach was implemented by python using SciPy, NumPy, and the non uniform rational B-spline surface (NURBS) python libraries. Figure 6(a) presents six steps in the optimization process of the paraboloid surface case, where each step is further divided into (a) and (b) these two sub-steps, corresponding to using the transmitter coil to optimize the receiver coil and using the optimized receiver coil to optimize the transmitter coil, respectively. In this case, we set the desired power to be transferred as 0.05 Watt to highlight the effects of each turn of the coils. With an Intel™ Core i5™ 3.7GHZ CPU, it took about 35 min to finish each step of the computing. In the figure, it can be found that with the development of the optimization process, the distance between the two coils was reducing, the sizes of both coils were shrinking as well, as only a few turns were enough to transfer the desired power at a shorter distance. In Fig. 6(b), the parameters of coils regarding each step are presented, where the length and the resistance of the coils decreased during the optimization process while the power that can be transmitted always met the requirement. In the final several steps, the power that can be transferred increased due to that the turn of the coil was set to be an integer. While the distance between two coils was small and the number of turns was low, an extra turn may lead to large changes of the power that can be transferred. It is worth mentioning that in the proposed approach, the effectiveness of the optimization results highly depends on the ROIs selected by the users and in general, the closer the ROIs of the transmitter and the receiver coils are, the higher the efficiency is. The initial position of the transmitter coil is specified at S_T(0.5, 0.5), and a better initial position based on the prior knowledge of the designer can also accelerate the optimization process.

To further verify the calculation results, we used the ANSYS™ Maxwell™ simulation tool to explore the properties of Eq. (14) at its optimal position. In the simulation, we fixed the transmitter coil but changed a of the center positions S_R(a, b) (along u-axis of the NURBS) from 0.35 to 0.65 with a 0.005 step and b (along v-axis) from 0.50 to 0.80 with a 0.005 step, resulting in 3721 setups. With the same computer, it took about 10 s to finish the simulation of a setup. Figure 7(a) presents the relations between those positions and the power that can be transmitted by the coils. It can be found that the maximum was around S_R(0.485, 0.735), which is the optimal position found by the algorithm. It is worth mentioning that the optimal position was not at the center of the figure due to the fact that the boundary of the coils reached the boundary of the ROIs. Figure 7(b) presents the magnetic B-field of the IPT system at the optimal position.

To explore the possibility of using the direct printing method to fabricate the coils, we redesigned and fabricated a pair of coils that can transmit 0.1 W power with a 5 Ω resistive load deployed on the receiver side. Figure 8(a) presents the design which was printed by an Ultimaker S5 printer with two grooves at the position of the designed coils. The height and width of them were set the same as the parameters of the optimized transmitter and receiver coils, respectively. Silver ink (density: 3700 kg/m³, volume resistivity: <3 × 10⁻⁷ Ω m) was filled in the groove to create the coils (Fig. 8(b)). In Fig. 8(c), the transmitter and the receiver coils were placed in the designed relative position to form an IPT system and in Fig. 8(d), an light-emitting diode (LED) was lighted using the power transferred by the system.

Table 1 presents the calculated, simulated, and measured properties of the coils presented in Fig. 8. In the table, T stands for the transmitter coil and R represents the receiver coil. Regarding the resistance, for the transmitter coil, it was measured as 0.9 Ω and for the receiver coil, the resistance was about 0.5 Ω. The calculated self-inductance of the optimized transmitter coil was 3.38 × 10⁻⁶ H, and the simulation result was 3.35 × 10⁻⁶ H while the calculated self-inductance of the optimized receiver coil was 1.39 × 10⁻⁶ H, and the simulation result was 1.41 × 10⁻⁶ H. The calculation results and the simulation results of the self-inductance of the transmitter coil were 9.8% and 10.6% smaller than the measurement results, respectively. And the calculation result of the self-inductance of the receiver coil was 12% smaller than the measurement result, while the simulation result was 10.8% smaller than the measurement result. At this position, the mutual-inductance of the calculation and the simulation between the optimized coils were 5.94 × 10⁻⁷ H and 6.01 × 10⁻⁷ H, respectively. The calculation result of the mutual-inductance was 13.8% smaller than the measurement result, while the simulation result was 12.8% smaller than the measurement result. The quality factors of coils were not high, mainly due to the large resistances introduced by the silver ink.

For both the transmitter coil and the receiver coil, errors between the calculation and the simulation results were small, which verified the effectiveness of the proposed approach. However, errors were slightly larger compared to the measurement results. This might be caused by the manual filling process of the silver ink, where the filled silver ink might not be uniformly distributed in the grooves.

To further verify the proposed approach, we designed the transmitter coil regarding a fixed receiver coil using both the proposed approach and the approach introduced by Tao et al. [38], which suggests to map a 2D design to the 3D ROI using UV mapping. To make the two designs comparable, we set the center of both coils at the same position and the power to be transferred as 0.1 W. Figure 9 presents the transmitter coils designed by the two approaches and Table 2 presents calculation results of both designs. Regarding the length of coils, the coil designed by the proposed approach was 0.230 m and the result of Tao’s coil was 0.277 m. The shorter length of coil (17%) suggested by the proposed approach reduced the cost of silver ink and might free more space in the ROI for other purposes. If both coils were manufactured with the same type of ink and had the same cross-sectional area, the resistances of them were linearly correlated to the lengths. Therefore, the resistance of the coil designed by the proposed approach was also 17% smaller. Though the coil was shorter, the power transmitted by the coil designed by the proposed approach was larger (0.112 W) than that of Tao’s approach (0.096 W).

To evaluate the potential of using the proposed 3D coil design method in design, we redesigned a shaver by integrating an 3D IPT system with the desired power to be transferred as 1 W. Figure 10(a) presents the original design of the shaver and holder, where two surfaces were specified as the ROIs for the transmitter coil and the receiver coil, respectively. In the design of the IPT, we set the voltage as 5 V and the load resistance as 5 Ω. Figure 10(b) presents the optimized 3D transmitter and the receiver coils. Here, the length of the coil was 0.17 m for both the transmitter coil and the receiver coil. Using this design, the IPT system was able to transmit the power of 1.15 W according to calculation. In Fig. 10(c), the simulation results of the 3D IPT are presented where the power can be transmitted was 1.21 W. The absolute error of the power transmitted between the calculation and simulation was 0.06 W (4.96%).

Table 3 presents the calculated and simulated result of the coils presented in Fig. 10. In the table, T and R represent the transmitter coil and the receiver coil, respectively. The calculated resistance of the optimized receiver coil was 1.40 × 10⁻¹ Ω and the simulation result was 1.43 × 10⁻¹ Ω while the calculated self-inductance of the optimized receiver coil was 1.42 × 10⁻¹ Ω, and the simulation result was 1.46 × 10⁻¹ Ω. Regarding the self-inductance, for the transmitter coil, the calculated result was 3.08 × 10⁻⁶ H and the simulated result was 3.15 × 10⁻⁶ H, and for the receiver coil, the calculated result was 3.39 × 10⁻⁶ H and the simulated result was 3.31 × 10⁻⁶ H. The calculation results were 2.46% smaller than the simulation results. And the calculation result of the self-inductance of the receiver coil was 2.48% bigger than the simulation result. At this position, the mutual-inductance of the calculation and the simulation between the optimized coils were 2.00 × 10⁻⁷ H and 2.07 × 10⁻⁷ H, respectively. The calculation result of the mutual-inductance was 3.25% smaller than the simulation result.

In this section, we present a case of integrating the IPT system into personalized product design using conformal printing techniques. The case was set to print a smart soft liner of a hand splint for measuring physiological parameters of the hand, e.g., the skin temperature. Figure 11(a) presents the design of the hand splint, which follows the 3D scan of the hand. As the liner is often made of soft materials and put in-between the splint and the skin, we took the shape of the hand and the forearm as the design space of the receiver coil. A flat table was taken as the design space of the transmitter coil and the relative position between two design spaces was set to be that the arm was put on the table with the palm facing upwards. With these two design spaces and by setting 0.1 W as the desired power to be transferred, the receiver coil was generated using the proposed method as shown in Fig. 11(b). Using LSCM, we mapped the 3D design to 2D and printed the coil on soft thermoplastic polyurethane (TPU) film using a Voltera electronics printer. The coil trace was printed by Voltera FLEX 2 ink using a 225-μm nozzle. The dispense height was set as 0.16 mm for the flexible TPU film. And the feed rate of the nozzle was set as 500 mm/min. After printing, the substrate with the printed trace was cured at 60 °C for 20 min. Figure 11(c) shows the printing process and in Fig. 11(d), the printed receiver coil is presented. In Fig. 11(e), the printed liner (with the receiver coil) was put in the hand splint. As this form, the self-inductance of the coil was measured as 1.01 × 10⁻⁶ H, which was 6.93% smaller than the calculation result (1.08 × 10⁻⁶ H). In Fig. 11(f), we demonstrated the effectiveness of the printed receiver coil by powering an LED. In this experiment, the error between the calculation and the measurement results was much smaller, mainly due to that the coil was digitally fabricated by the electronics printer.

In this paper, we presented a computational design approach for designing 3D coils for IPT. Compared to the traditional design method, the coils are designed and optimized in 3D directly with the consideration of the electromagnetic constraints. Simulation and experiment results show that the proposed approach can be a useful method in minimizing the size as well as reducing the use of materials, both contribute to a more efficient design.

Current research is focused towards two directions of printing the designed coils for different applications. In the first direction, we are working on multi-layer conformal printing using thermoforming [49] to increase the power that can be transferred. At the second direction, we are exploring the possibilities of directly printing the 3D coils using 3D printed electronics.

The presented research was partially funded by the European Institute of Technology (EIT) AddMANU project and Dutch NWO Next UPPS—Integrated Design Methodology for Ultra Personalised Products and Services project. The work of Mr. Jun Xu was supported by the China Scholarship Council (CSC, No. 201806890022). The author would also like to express their appreciation to Mr. Rick van Veen, Mr. Martin Verwaal, Mr. Adrie Kooijman, Mr. Joris van Dam, and Ms. Tessa Essers for the fruitful discussions and kind assistance during the experiment.